WASHINGTON — Sen. John Cornyn raised the specter last week that illegal immigrants are pouring across the Mexican border — 300 per night in one spot, a rate that would command high-level attention in Washington and probably steer the debate over immigration policy and border security.

“Friend on border sez 300 ppl coming across his property every night. And Napolitano sez border is under control?” Cornyn wrote on his Twitter account, referring to Homeland Security Secretary Janet Napolitano.

Cornyn aides have declined for several days to identity the senator’s friend or his location or provide any other substantiation. The senator has more than 37,000 followers on the social media service, and dozens have helped spread the comment.

Such a border hot spot would see nearly 110,000 crossings per year, nearly a third of the number of people caught last year by federal agents along the entire 2,000-mile border.

Officials at the U.S. Customs and Border Protection and its parent agency, the Department of Homeland Security, declined to comment on Cornyn’s assertion or discuss how authorities would respond if they knew about a hot spot of the sort described.

The state of border security has been at the forefront of a debate about overhauling the nation’s immigration system. Cornyn and other conservatives insist that the border is not yet secure enough to consider legal status for the 11 million people in this country illegally.

Democrats and advocates for immigrants say the border has never been more secure. They accuse opponents of stalling.

U.S. Customs and Border Protection stopped about 365,000 people trying to cross into the U.S. in 2012 without permission, almost half of them in Texas. The number of people apprehended has fallen nearly every year since 2005, as Border Protection has beefed up its budget and more than doubled the number of Border Patrol agents to more than 21,000, an all-time high, since 2004.

While the number of apprehensions in South Texas rose slightly last year, the trend remains downward in other border states, “reflecting fewer individuals crossing the border,” said Michael Friel, a spokesman for the agency.

That doesn’t mean the border is sealed, said Donald Reay, executive director of the Texas Border Sheriffs Coalition.

“I don’t think we really know how many [people] are trying to cross,” he said.

Reay said he hasn’t heard of a situation as egregious as the one Cornyn tweeted about but said he trusts the senator and wouldn’t call it impossible.

Federal officials and border security experts said that without a location, they cannot judge the validity of the senator’s claim.

What would happen if there was such a large influx of immigrants, trudging through someone’s property in the dead of night?

It’s unclear, because of staffing issues and evolving law enforcement tactics.

“It’s going to vary from place to place, and that’s the reality,” said Christopher Bauder, president of the National Border Patrol Council, a union that represents more than 17,000 agents and staff.

Susan Long, co-director of Syracuse University’s Transactional Records Access Clearinghouse, a data research organization, said all available data suggests that fewer migrants are trying to cross the border. She called a claim of 300 crossing per night in one place “inconsistent with all of the other — albeit not terribly wonderful — information about people crossing that aren’t caught.”